A new result on Chvátal's conjecture
Let \(I\) be an ideal of the finite set \(X\). An element \(j\in X\) is a dominant element of \(I\), if \(j\notin A\in I\) implies \(A\backslash\{a\}\cup\{j\}\in I\) for every \(a\in A\). (This is a generalization of ideals equiped by a hereditary tree structure, introduced by \textit{D.-L. Wang} and \textit{P. Wang} [Discrete Math. 24, 95- 101 (1978; Zbl 0384.05003)].) The paper's main result shows that Chvátal's conjecture [Proc. 1st Working Sem. Hypergraphs, Columbus 1972, Lect. Notes Math. 411, 61-66 (1974; Zbl 0305.05128)] holds for ideals having dominant elements. The paper also proves that if the bases of the ideal \(I\) can be partitioned into two strong delta systems with one-element kernels, then the ideal \(I\) satisfies Chvátal's conjecture.
scientific article; zbMATH DE number 68491
scientific article